Page:A History Of Mathematical Notations Vol I (1928).djvu/421

IV
SYMBOLS IN GEOMETRY
(ELEMENTARY PART)

A. ORDINARY ELEMENTARY GEOMETRY

357. The symbols sometimes used in geometry may be grouped roughly under three heads: (1) pictographs or pictures representing geometrical concepts, as △ representing a triangle; (2) ideographs designed especially for geometry, as ~ for “similar”; (3) symbols of elementary algebra, like + and −.

Early use of pictographs.—The use of geometrical drawings goes back at least to the time of Ahmes, but the employment of pictographs in the place of words is first found in Heron’s Dioptra. Heron (150 A.D.) wrote ▷ for triangle, ${\stackrel {\mathrm {o} \upsilon }{=}}$ for parallel and parallelogram, also ${\stackrel {\rho {\text{'}}}{=}}$ for parallelogram, ▭‵ for rectangle, ⊙̌ for circle.^[1] Similarly, Pappus (fourth century A.D.) writes ○ and ⊙ for circle, ▽ and △ for triangle, ∟ for right angle, ${\stackrel {\mathrm {o} \iota }{=}}$ or $=$ for parallel, □ for square.^[2] But these were very exceptional uses not regularly adopted by the authors and occur in few manuscripts only. They were not generally known and are not encountered in other mathematical writers for about one thousand years. Paul Tannery calls attention to the use of the symbol □ in a medieval manuscript to represent, not a square foot, but a cubic foot; Tannery remarks that this is in accordance with the ancient practice of the Romans.^[3] This use of the square is found in the Triparty of Chuquet (§ 132) and in the arithmetic of De la Roche.

358. Geometric figures were used in astrology to indicate roughly the relative positions of two heavenly bodies with respect to an observer. Thus ☌, ☍, □, △, ⚹ designated,^[4] respectively, conjunction,

1. Notices et extraits des manuscrits de la Bibliothèque impériale, Vol. XIX, Part II (Paris 1858), p. 173.
2. Pappi Alexandrini Collectionis quae supersunt (ed. F. Hultsch), Vol. III, Tome I (Berlin, 1878), p. 126–31.
3. Paul Tannery, Mémoires scientifiques, Vol. V (Toulouse and Paris, 1922), p. 73.
4. Kepler says: “Quot, sunt igitur aspectus? Vetus astrologia agnoscit tantum quinque: conjunctionem (☌), cum radii planetarum binorum in Terram descendentes in unam conjunguntur lineam; quod est veluti principium aspectuum omnium. 2) Oppositionem (☍), cum bini radii sunt ejusdem rectae partes, seu

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